The pentagonal theorem of sixty-three and generalizations of Cauchy's lemma

In this article, we study the representability of integers as sums of pentagonal numbers, where a pentagonal number is an integer of the form $P_5(x)=\frac{3x^2-x}{2}$ for some non-negative integer $x$. In particular, we prove the "pentagonal theorem of $63$", which states that a sum of pentagonal numbers represents every non-negative integer if and only if it represents the integers $1$, $2$, $3$, $4$, $6$, $7$, $8$, $9$, $11$, $13$, $14$, $17$, $18$, $19$, $23$, $28$, $31$, $33$, $34$, $39$, $42$, and $63$. We also introduce a method to obtain a generalized version of Cauchy's lemma using representations of binary integral quadratic forms by quaternary quadratic forms, which plays a crucial role in proving the results.